We define data transformations that leave certain classes of distributions invariant, while acting in a specific manner upon the parameters of the said distributions. It is shown that under such transformations the maximum likelihood estimators behave in exactly the same way as the parameters being estimated. As a consequence goodness--of--fit tests based on standardized data obtained through the inverse of this invariant data--transformation reduce to the case of testing a standard member of the family with fixed parameter values. While presenting our results, we also provide a selective review of the subject of equivariant estimators always in connection to invariant goodness--of--fit tests. A small Monte Carlo study is presented for the special case of testing for the Weibull distribution, along with real--data illustrations.

翻译：我们定义了某些使特定分布族保持不变、同时以特定方式作用于所述分布参数的数据变换。研究表明，在此类变换下，最大似然估计量的行为与被估计参数完全一致。因此，基于通过该不变性数据变换的逆变换获得的标准化数据的拟合优度检验，可简化为对具有固定参数值族中标准成员进行检验的情形。在呈现结果的同时，我们始终结合不变性拟合优度检验，对等变性估计量这一主题进行了选择性综述。针对威布尔分布检验的特例，我们开展了小型蒙特卡罗研究，并辅以真实数据示例进行说明。